Perform the row operation, $R_3+2R_1\rightarrow R_3$, on the following matrix. $\left[\begin{array} {ccc} 1 & 6 & 9 & 3 \\ 2 & 5 & 8 & 1 \\ 0 & 2 & 2 & 4 \end{array} \right] $
Solution: Background There are three basic row operations that can be performed on matrices. $R_i \leftrightarrow R_j$. This symbol tells us to interchange rows $i$ and $j$. $cR_i \rightarrow R_i$. This symbol tells us to multiply a row $i$ by a constant $c$. $R_i + cR_j \rightarrow R_i$. This symbol tells us to add $c$ times row $j$ to row $i$. Finding the new row to be used For the given matrix, $R_1$ and $R_3$ are given below. $R_1=\left[\begin{array} {ccc} 1 & 6 & 9 & 3 \end{array} \right] ~~~~~ R_3=\left[\begin{array} {ccc} 0 & 2 & 2 & 4 \end{array} \right]$ We are asked to perform the row operation, $R_3+2R_1\rightarrow R_3$. Therefore, we must add $2R_1$ to $R_3$. $\begin{aligned}R_3+2R_1 &= \left[\begin{array} {ccc} 0 & 2 & 2 & 4 \end{array} \right] + 2\left[\begin{array} {ccc} 1 & 6 & 9 & 3 \end{array} \right] \\\\&=\left[\begin{array} {ccc} 2 & 14 & 20 & 10 \end{array} \right]\end{aligned}$ Substituting the row Now, we must substitute row $R_3$ with $R_3+2R_1$. $\left[\begin{array} {ccc} 1 & 6 & 9 & 3 \\ 2 & 5 & 8 & 1 \\ {0} & {2} & {2} & {4} \end{array} \right] \xrightarrow{R_3+2R_1\rightarrow R_3} \left[\begin{array} {ccc} 1 & 6 & 9 & 3 \\ 2 & 5 & 8 & 1 \\ {2} & {14} & {20} & {10} \end{array} \right]$ Summary Our resultant matrix is the following. $\left[\begin{array} {ccc} 1 & 6 & 9 & 3 \\ 2 & 5 & 8 & 1 \\ 2 & 14 & 20 & 10 \end{array} \right]$